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%I #20 Aug 18 2019 17:25:36
%S 0,0,1,0,1,0,0,1,1,0,0,1,2,1,0,0,1,3,3,1,1,0,1,4,6,4,2,1,0,1,5,10,10,
%T 6,3,1,0,1,6,15,20,16,9,4,1,0,1,7,21,35,36,25,13,5,2,0,1,8,28,56,71,
%U 61,38,18,7,3,0,1,9,36,84,127,132,99,56,25,10,4,0,1,10,45,120,211,259,231,155,81,35,14,5
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where sequence a_k of column k is the expansion of x/((1 - x - x^4)*(1 - x)^(k - 1)).
%C Each row sequence a_n (for n > 0) is produced by a polynomial of degree n-1, whose (rational) coefficients are given in row n of A145140/A145141. The coefficients *(n-1)! are given in A145142.
%C Each column sequence a_k is produced by a recursion, whose coefficients are given by row k of A145152.
%H Alois P. Heinz, <a href="/A145153/b145153.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: x/((1-x-x^4)*(1-x)^(k-1)).
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, ...
%e 0, 1, 3, 6, 10, 15, 21, ...
%e 0, 1, 4, 10, 20, 35, 56, ...
%e 1, 2, 6, 16, 36, 71, 127, ...
%p A:= proc(n, k) coeftayl (x/ (1-x-x^4)/ (1-x)^(k-1), x=0, n) end:
%p seq(seq(A(n, d-n), n=0..d), d=0..13);
%t a[n_, k_] := SeriesCoefficient[x/(1 - x - x^4)/(1 - x)^(k - 1), {x, 0, n}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Dec 05 2013 *)
%Y Rows 0-9 give: A000004, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.
%Y Columns 0-9 give: A017898(n-1) for n>0, A003269, A098578, A145131, A145132, A145133, A145134, A145135, A145136, A145137.
%Y Main diagonal gives: A145138.
%Y Antidiaginal sums give: A145139.
%Y Numerators/denominators of polynomials for rows give: A145140/A145141.
%Y Cf. A145142, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150, A145151, A145152.
%K nonn,tabl
%O 0,13
%A _Alois P. Heinz_, Oct 03 2008
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